We’re taught that exponents are repeated multiplication. This is a good intro-duction, but it breaks down on 31.5 and the brain-twisting 00. How do you repeat zero “zero” times and get 1 — without melting your brain?
You can’t, not while exponents are repeated multiplication. Today our men-tal model is due for an upgrade.
10.1 Viewing Arithmetic As Transformations
Let’s step back — how do we learn arithmetic? We’re taught that numbers are counts of something (fingers), addition is combining those counts (3 + 4 = 7) and multiplication is repeated addition (2 × 3 = 2 + 2 + 2 = 6).
This interpretation works for round numbers like 2 and 10. Strange con-cepts like -1 andp2seem to fit. Why?
Our model was incomplete. Numbers aren’t just a count; a better view-point is a position on a line. This position can be negative (-1), between other numbers (p2), or in another dimension (i).
Arithmetic became a general way to transform a number. Addition is sliding along the number line (+3 means slide 3 to the right) and multiplication is scaling (×3means scale it up 3x).
So what are exponents?
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10.2 Enter the Expand-o-tron
Let me introduce the Expand-o-tron 3000.
Yes, this device looks like a shoddy microwave — but instead of heating food, it grows numbers. Put a number in and a new one comes out. Here’s how:
• Start with 1.0
• Set the growth to the desired change after one second (2x, 3x, 10.3x)
• Set the time to the number of seconds to grow
• Push start
And shazam! The bell rings and we pull out our shiny new number. Sup-pose we want to change 1.0 into 9:
• Put 1.0 in the expand-o-tron
• Set the change for “3x” growth, and the time for 2 seconds
• Push start
The number starts transforming as soon as we begin: We see 1.0, 1.1, 1.2. . . and just as we finish the first second, we’re at 3.0. And the growth continues: 3.1, 3.5, 4.0, 6.0, 7.5. . . And just as the 2nd second ends we’re at 9.0. Behold our shiny new number!
Mathematically, the expand-o-tron (exponent function) does this:
or i g i nal · g r ow thd ur at i on
= new
CHAPTER 10. UNDERSTANDING EXPONENTS 75
or g r ow t hd ur at i on
= new
or i g i nal
For example, 32 = 9/1. The base is the amount to grow each unit (3x), and the exponent is the amount of time (2). A formula like2n means “Use the expand-o-tron at 2x growth for n seconds”.
Remember, we always start with 1.0 in the expand-o-tron to see how it changes a single unit. If we want to see what would happen if we started with 3.0 in the expand-o-tron, we just scale up the final result. For example:
• “Start with 1 and double 3 times” means1 · 23= 1 · 2 · 2 · 2 = 8
• “Start with 3 and double 3 times” means3 · 23= 3 · 2 · 2 · 2 = 24
Whenever you see an plain exponent by itself (like 23), we’re implicitly starting with 1.0 and transforming with 2x growth for 3 seconds.
10.3 Understanding the Exponential Scaling Factor
When multiplying, we can just state the final scaling factor. Want it 8 times larger? Multiply by 8. Done.
Exponents are a bit. . . finicky:
You: I’d like to grow this number.
Expand-o-tron: Ok, stick it in.
You: How big will it get?
Expand-o-tron: Gee, I dunno. Let’s find out. . . You: Find out? I was hoping you’d
kn-Expand-o-tron: Shh!!! It’s growing! It’s growing!
You: . . .
Expand-o-tron: It’s done! My masterpiece is alive!
You: Can I go now?
The expand-o-tron is indirect. Just looking at it, you’re not sure what it’ll do: What does310 mean to you? How does it make you feel? Instead of a nice finished scaling factor, exponents want us to feel, relive, even smell the growing process. Whatever you end with is your scaling factor.
It sounds roundabout and annoying. You know why? Most things in na-ture don’t know where they’ll end up!
Do you think bacteria plans on doubling every 14 hours? No — it just eats the moldy bread you forgot about in the fridge as fast as it can, and as it gets more the blob starts growing even faster (a purely hypothetical situation, of course). To predict the behavior, we input how fast they’re growing (current rate) and how long they’ll be changing (time) to work out their final value.
The answer has to be worked out — exponents are a way of saying “Begin with these conditions, start changing, and see where you end up”. The expand-o-tron (or our calculator) does the work by crunching the numbers to get the final scaling factor. But someone has to do it.
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10.4 Understanding Fractional Powers
Let’s see if the expand-o-tron can help us understand exponents. First up: what does at21.5mean?
It’s confusing when we think of repeated multiplication. But the expand-o-tron makes it simple: 1.5 is just the amount of time in the machine.
• 21means 1 second in the machine (2x growth)
• 22means 2 seconds in the machine (4x growth)
So21.5means 1.5 seconds in the machine, so somewhere between 2x and 4x growth. The idea of “repeated counting” had us stuck with integers.
10.5 Multiplying Exponents
What if we want two growth cycles back-to-back? Let’s say we use the machine for 2 seconds, and then use it for 3 seconds at the exact same power:
x2· x3=?
Think about your regular microwave — isn’t this the same as one continu-ous cycle of 5 seconds? It sure is. As long as the power setting (base) stayed the same, we can just add the time:
xy· xz= xy+z
Again, the expand-o-tron gives us a scaling factor to change our number.
To get the total effect from two consecutive uses, we just multiply the scaling factors together.
10.6 Square Roots
Let’s keep going. Let’s say we’re at power level a and grow for 3 seconds:
a3
Not too bad. Now what would growing for half that time look like? It’d be 1.5 seconds:
a1.5
Now what would happen if we did that twice?
a1.5· a1.5= a3
Said another way: partial growth×partial growth = full growth
Looking at this equation, we see “partial growth” is the square root of full growth! If we divide the time in half we get the square root scaling factor. And if we divide the time in thirds?
a1· a1· a1= a3
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Or: partial growth×partial growth×partial growth = full growth
And we get the cube root! For me, this is an intuitive reason why divid-ing the exponents gives roots: we split the time into equal amounts, so each
“partial growth” period must have the same effect. If three identical effects are multiplied together, it means they’re each a cube root.
10.7 Negative Exponents
Now we’re on a roll — what does a negative exponent mean? Well, “negative seconds” means going back in time! If going forward grows by a scaling factor, going backwards should shrink by it.
2−1= 1 21
The sentence means “1 second ago, we were at half our current amount (aka 1/21)”. In fact, this is a neat part of any exponential graph, like2x:
Pick a point like 3.5 seconds (23.5= 11.3). One second in the future we’ll be at double our current amount (24.5= 22.5). One second ago we were at half our amount (22.5= 5.65).
This works for any number! Wherever 1 million is in our doubling growth curve, we were at 500,000 one second before it.
10.8 Taking the Zeroth Power
Now let’s try the tricky stuff: what does30mean? Well, we set the machine for 3x growth, and use it for. . . zero seconds. Zero seconds means we don’t even use the machine!
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Our new and old values are the same (new = old), so the scaling factor is 1. Using 0 as the time (power) means there’s no change at all. The scaling factor is always 1.
10.9 Taking Zero As a Base
How do we interpret0x? Well, our growth amount is “0x” — after a second, the expand-o-tron obliterates the number and turns it to zero. But if we’ve obliterated the number after 1 second, it really means any amount of time will destroy the number:
01/n= nth root of01= nth root of 0 = 0
No matter the tiny power we raise it to, it will be some root of 0.
10.10 Zero to The Zeroth Power
At last, the dreaded00. What does it mean? The expand-o-tron to the rescue:
00means a 0x growth for 0 seconds!
Although we planned on obliterating the number, we never used the ma-chine. No usage means new = old, and the scaling factor is just 1.00= 1 · 00= 1 · 1 = 1— it doesn’t change our original number. Mystery solved!
(For the math geeks: Defining00as 1 makes many theorems work smoothly.
In reality, 00 depends on the scenario (continuous or discrete) and is under debate. The microwave analogy isn’t about rigor: it helps us see why00= 1can be reasonable, in a way that “repeated counting” does not.)
10.11 Advanced: Repeated Exponents (a to the b to the c)
Repeated exponents are tricky. What does this mean?
(2a)b
It’s “repeated multiplication, repeated” — another way of saying “do that exponent thing once, and do it again”. Let’s dissect it:
(23)4
• First, I want to grow by doubling each second: do that for 3 seconds (23)
• Then, whatever my number is (8x), I want to grow by that new amount for 4 seconds (84)
The first exponent (3) just knows to take “2” and grow it by itself 3 times.
The next exponent (4) just knows to take the previous amount (8) and grow
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it by itself 4 times. Each time unit in “Phase II” is the same as repeating all of Phase I:
(23)4= 23× 23× 23× 23= 23+3+3+3= 212
Repeated counting helps us get our bearings. But then we bring out the expand-o-tron analogy: we grow for 3 seconds in Phase I, and redo that for 4 more seconds in Phase II. The expand-o-tron works for fractional powers:
(23.1)4.2
which means “Grow for 3.1 seconds, and use that new growth rate for 4.2 seconds”. We can smush together the time (3.1×4.2) like this:
(ab)c= ab·c= (ac)b
Repeated exponents is a bit strange, so try some examples:
• (21)x means “Grow at 2 for 1 second, and ‘do that growth’ for x more seconds”.
• 7 = (70.5)2 means “We can jump to 7 all at once. Or, we can plan on growing to 7 but only use half the time (p7). But we can do that process for 2 seconds, which gives us the full amount (p7squared = 7).”
We’re like kids learning that3 × 7 = 7 × 3.
10.12 Advanced: Rewriting Exponents For The Grower
The expand-o-tron is a bit strange: numbers start growing the instant they’re inside, but we specify the desired growth at the end of each second.
We say we want 2x growth at the end of the first second. But how do we know what rate to start off with? How far along should we be at 0.5 seconds?
It can’t be the full amount, or else we’ll overshoot our goal as our interest compounds.
Here’s the key: Growth curves written like 2x are from the observer’s viewpoint, not the grower.
The value “2” is measured at the end of the interval and we work backwards to create the exponent (Oh, it looks like you’re growing at2x). This is conve-nient for us, but not the growing quantity — bacteria, radioactive elements and money don’t care about lining up with our ending intervals!
No, these critters know their current, instantaneous growth rate, and don’t try to line up their final amounts with our boundaries. It’s just like understand-ing radians vs. degrees — radians are “natural” because they are measured from the mover’s viewpoint.
To get into the grower’s viewpoint, we use themagical number e. There’s much more to say, but we can convert any “observer-focused” formula like2x into a “grower-focused” one:
2x= (el n(2))x= el n(2)x
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In this case, ln(2) = .693 = 69.3% is the instantaneous growth rate needed to look like2x to an observer. When you ask for “2x growth at the end of each period”, the expand-o-tron knows this means to grow the number at a rate of 69.3%.
There’s more details, but remember this:
• The instantaneous growth rate controlled by the bacteria
• The overall rate measured at the end of each interval by the observer Underneath it all, every exponential curve is just a scaled version ofex:
ax= (el n(a))x= el n(a)x
Every exponent is a variation of e, just like every number is a scaled version of 1.
10.13 Why Use This Analogy?
Does the expand-o-tron exist? Do numbers really gather up in a number line?
Nope — they’re ways of looking at the world.
The expand-o-tron removes the mental hiccups when seeing21.5or even00. Everything from slide rules toEuler’s Formulabegins to click once we recognize the core theme of growth — even beasts likeii can be tamed.
Friends don’t let friends think of exponents as repeated multiplication.
Happy math.